Two-dimensional Periodic Flow Research Articles

Two-Dimensional Surface Periodic Flows of an Incompressible Fluid in Various Models of the Medium

Two-Dimensional Surface Periodic Flows of an Incompressible Fluid in Various Models of the Medium

Multiscaled inviscid Taylor-Green vortex flow for examining energy conservation error in incompressible flows

This study investigates kinetic energy conservation errors in incompressible flows using multiplexed inviscid Taylor-Green vortices. Fourth-order precision Runge-Kutta methods, specifically five- and six-step methods, were employed and an inviscid two-dimensional periodic flow field was used as a test case. This method allows the energy conservation error to be suppressed to a very low level. Recent previous studies have dealt with turbulence generation using a multi-scale turbulence grid. In this study, the Taylor-Green vortex of the analytical solution is multiplexed to generate a flow field with a number of wavenumbers. Visualisation results of the initial field and the energy distribution over time were obtained. It was observed that the errors in the time evolution of the fluid energy were extremely small, and that the errors actually increased with time. However, the influence of the time step range was limited and the analytical and numerical solutions were in general agreement. As a result, it was confirmed that the inviscid Taylor-Green vortex is effective in examining the energy conservation error for incompressible flows.

An Elliptic Blending Turbulence Model-Based Scale-Adaptive Simulation Model Applied to Fluid Flows Separated from Curved Surfaces

On the basis of a previously developed elliptic blending turbulence model (SST–k–ω–φ–α model), a scale-adaptive simulation (SAS) model is developed by following Menter and Egorov’s SAS concept. An SAS source term, which is related to the ratio of the modeled turbulence scale to the von Kármán length scale, is introduced into the corresponding length-scale determining equation. The major motivation of this study is that the conventional unsteady Reynolds-averaged Navier–Stokes (URANS) models provide only large-scale unsteadiness. The introduction of the SAS term allows the proposed SAS model to dynamically adjust to resolved structures in a URANS framework because this term is sensitive to resolved fluctuations. The predictive capabilities of the proposed SAS model are demonstrated by computing the complex flow configurations in three cases with flow separation from curved surfaces, namely, three-dimensional (3D) diffuser flow, two-dimensional (2D) periodic hills flow, and 2D U-turn duct flow. For comparison, the results predicted by the SST–k–ω–φ–α model and the Menter and Egorov’s SAS model (SST–SAS) are provided. The results are also compared with the relevant experimental, direct numerical simulation, and large eddy simulation data. The results show that the SST–k–ω–φ–α model cannot capture the critical features for all three flows, and that the SST–SAS model is able to predict the results reasonably well. The proposed SAS model is capable of resolving more portions of the turbulence structures, and it yields the best results in all the cases.

Phase-synchronization properties of laminar cylinder wake for periodic external forcings

Abstract

Analytical investigation of the vertical structure of periodic flow

Analytical solution to the vertical two-dimensional periodic flow in a rectangular basin is presented in this paper. This solution takes into account the effects of the bottom friction through a slip condition, the horizontal turbulent viscosity, and the vertical turbulent viscosity. These induce a better evaluation of the free surface elevation and velocity.The introduction of a slip condition at the bottom considerably modifies the phase for the free surface elevation and especially the amplitude for the velocity at the bottom layer whereas the introduction of the horizontal turbulent viscosity considerably modifies the phase for the free surface elevation.The model proposed here is more relevant for value on the bottom friction coefficient ranging from 10−4 ms−1 to 10−1 ms−1. Within this range, the slip condition at the bottom is really effective and an ideal value on the bottom friction coefficient for the slip condition is as much as 10−2 ms−1.The analytical solution confirms the existence of a large difference in phase between the bottom stress and the depth-averaged velocity and that this difference increases with the bottom friction. This leads to warn of using the depth-averaged velocity for calculating the bottom stress.

On Langmuir circulation in 1 : 2 and 1 : 3 resonance

This paper is concerned with the nonlinear dynamics of spanwise periodic longitudinal vortex modes (Langmuir circulation (LC)) that arise through the instability of two-dimensional periodic flows (waves) in a non-stratified uniformly sheared layer of finite depth. Of particular interest is the excitation of the vortex modes either in the absence of interaction or in resonance, as described by nonlinear amplitude equations built upon the mean field Craik–Leibovich (CL) equations. Since Y-junctions in the surface footprints of Langmuir circulation indicate sporadic increases (doubling) in spacing as they evolve to the scale of sports stadiums, interest is focused on bifurcations that instigate such changes. To that end, surface patterns arising from the linear and nonlinear excitation of the vortex modes are explored, subject to two parameters: a Rayleigh number ${\mathcal{R}}$ present in the CL equations and a symmetry breaking parameter $\unicode[STIX]{x1D6FE}$ in the mixed free surface boundary conditions that relax to those at the layer bottom where $\unicode[STIX]{x1D6FE}=0$. Looking first to linear instability, it is found as $\unicode[STIX]{x1D6FE}$ increases from zero to unity, that the neutral curves evolve from asymmetric near onset to almost symmetric. The nonlinear dynamics of single modes is then studied via an amplitude equation of Ginzburg–Landau type. While typically of cubic order when the bifurcation is supercritical (as it is here) and the neutral curves are parabolic, the Ginzburg–Landau equation must instead here be of quartic order to recover the asymmetry in the neutral curves. This equation is then subjected to an Eckhaus instability analysis, which indicates that linearly unstable subharmonics mostly reside outside the Eckhaus boundary, thereby excluding them as candidates for excitation. The surface pattern is then largely unchanged from its linear counterpart, although the character of the pattern does change when $\unicode[STIX]{x1D6FE}\ll 1$ as a result of symmetry breaking. Attention is then turned to strong resonance between the least stable linear mode and a sub-harmonic of it, as described by coupled nonlinear amplitude equations of Stuart-Landau type. Both 1 : 2 and 1 : 3 resonant interactions are considered. Phase plots and bifurcation diagrams are employed to reveal classes of solution that can occur. Dominant over much of the ${\mathcal{R}}$-$\unicode[STIX]{x1D6FE}$ range considered are non-travelling pure- and mixed-mode equilibrium solutions that act singly or together. To wit, pure modes solutions alone act to realise windrows with spacings in accord with linear theory, while bistability can realise Y-junctions and, depending upon initial conditions, double or even triple the dominant spacing of LC.

A Lagrangian criterion of unsteady flow separation for two-dimensional periodic flows

The present paper proposes a Lagrangian criterion of unsteady flow separation for two-dimensional periodic flows based on the principle of weighted averaging zero skin-friction given by Haller (HALLER, G. Exact theory of unsteady separation for two-dimensional flows. Journal of Fluid Mechanics, 512, 257–311 (2004)). By analyzing the distribution of the finite-time Lyapunov exponent (FTLE) along the no-slip wall, it can be found that the periodic separation takes place at the point of the zero FTLE. This new criterion is verified with an analytical solution of the separation bubble and a numerical simulation of lid-driven cavity flows.

Two- and three-dimensional wake transitions of an impulsively started uniformly rolling circular cylinder

This paper presents the characteristics of the different stages in the evolution of the wake of a circular cylinder rolling without slipping along a wall at constant speed, acquired through numerical stability analysis and two- and three-dimensional numerical simulations. Reynolds numbers between 30 and 300 are considered. Of importance in this study is the transition to three-dimensionality from the underlying two-dimensional periodic flow and, in particular, the way that the associated transitions influence the fluid forces exerted on the cylinder and the development and the structure of the wake. It is found that the steady two-dimensional flow becomes unstable to three-dimensional perturbations at $Re_{c,3D}=37$, and that the transition to unsteady two-dimensional flow – or periodic vortex shedding – occurs at $Re_{c,2D}=88$, thus validating and refining the results of Stewart et al. (J. Fluid Mech. vol. 648, 2010, pp. 225–256). The main focus here is on Reynolds numbers beyond the transition to unsteady flow at $Re_{c,2D}=88$. From impulsive start up, the wake almost immediately undergoes transition to a periodic two-dimensional wake state, which, in turn, is three-dimensionally unstable. Thus, the previous three-dimensional stability analysis based on the two-dimensional steady flow provides only an element of the full story. Floquet analysis based on the periodic two-dimensional flow was undertaken and new three-dimensional instability modes were revealed. The results suggest that an impulsively started cylinder rolling along a surface at constant velocity for $Re\gtrsim 90$ will result in the rapid development of a periodic two-dimensional wake that will be maintained for a considerable time prior to the wake undergoing three-dimensional transition. Of interest, the mean lift and drag coefficients obtained from full three-dimensional simulations match predictions from two-dimensional simulations to within a few per cent.

Retraction Note to: Generation of topological chaos in the stokes flow in a rectangular cavity

We have investigated the two-dimensional periodic flow of a viscous incompressible fluid in the Stokes approximation inside a rectangular cavity, which is driven by motion of the top and bottom boundaries. We have obtained the analytical solution of this problem by the improved reduction method for a given distribution of the velocities of motion of the boundaries. We have detected the existence of three periodic points in the flow that generate the chaotic motion of a selected passive fluid. The origin of topological chaos in the Stokes flow without stirrers placed inside the cavity has been shown.

Low-Reynolds-number wakes of elliptical cylinders: from the circular cylinder to the normal flat plate

AbstractWhile the wake of a circular cylinder and, to a lesser extent, the normal flat plate have been studied in considerable detail, the wakes of elliptic cylinders have not received similar attention. However, the wakes from the first two bodies have considerably different characteristics, in terms of three-dimensional transition modes, and near- and far-wake structure. This paper focuses on elliptic cylinders, which span these two disparate cases. The Strouhal number and drag coefficient variations with Reynolds number are documented for the two-dimensional shedding regime. There are considerable differences from the standard circular cylinder curve. The different three-dimensional transition modes are also examined using Floquet stability analysis based on computed two-dimensional periodic base flows. As the cylinder aspect ratio (major to minor axis) is decreased, mode A is no longer unstable for aspect ratios below 0.25, as the wake deviates further from the standard Bénard–von Kármán state. For still smaller aspect ratios, another three-dimensional quasi-periodic mode becomes unstable, leading to a different transition scenario. Interestingly, for the 0.25 aspect ratio case, mode A restabilises above a Reynolds number of approximately 125, allowing the wake to return to a two-dimensional state, at least in the near wake. For the flat plate, three-dimensional simulations show that the shift in the Strouhal number from the two-dimensional value is gradual with Reynolds number, unlike the situation for the circular cylinder wake once mode A shedding develops. Dynamic mode decomposition is used to characterise the spatially evolving character of the wake as it undergoes transition from the primary Bénard–von Kármán-like near wake into a two-layered wake, through to a secondary Bénard–von Kármán-like wake further downstream, which in turn develops an even longer wavelength unsteadiness. It is also used to examine the differences in the two- and three-dimensional near-wake state, showing the increasing distortion of the two-dimensional rollers as the Reynolds number is increased.

Dispersion in the large-deviation regime. Part 2. Cellular flow at large Péclet number

AbstractA standard model for the study of scalar dispersion through the combined effect of advection and molecular diffusion is a two-dimensional periodic flow with closed streamlines inside periodic cells. Over long time scales, the dispersion of a scalar released in this flow can be characterized by an effective diffusivity that is a factor$\mathit{Pe}^{1/2}$larger than molecular diffusivity when the Péclet number$\mathit{Pe}$is large. Here we provide a more complete description of dispersion in this regime by applying the large-deviation theory developed in Part 1 of this paper. Specifically, we derive approximations to the rate function governing the scalar concentration at large time$t$by carrying out an asymptotic analysis of the relevant family of eigenvalue problems. We identify two asymptotic regimes and, for each, make predictions for the rate function and spatial structure of the scalar. Regime I applies to distances$|\boldsymbol {x}|$from the scalar release point that satisfy$|\boldsymbol {x}|= O(\mathit{Pe}^{1/4} t)$. The concentration in this regime is isotropic at large scales, is uniform along streamlines within each cell, and varies rapidly in boundary layers surrounding the separatrices between adjacent cells. The results of homogenization theory, yielding the$O(\mathit{Pe}^{1/2})$effective diffusivity, are recovered from our analysis in the limit$|\boldsymbol {x}|\ll \mathit{Pe}^{1/4} t$. Regime II applies when$|\boldsymbol {x}|=O(\mathit{Pe}\, t/{\rm log}\, \mathit{Pe})$and is characterized by an anisotropic concentration distribution that is localized around the separatrices. A novel feature of this regime is the crucial role played by the dynamics near the hyperbolic stagnation points. A consequence is that in part of the regime the dispersion can be interpreted as resulting from a random walk on the lattice of stagnation points. The two regimes overlap so that our asymptotic results describe the scalar concentration over a large range of distances$|\boldsymbol {x}|$. They are verified against numerical solutions of the family of eigenvalue problems yielding the rate function.

Asymptotic Growth Rates and Strong Bending of Turbulent Flame Speeds of G-Equation in Steady Two-Dimensional Incompressible Periodic Flows

The study of turbulent flame speeds (the large time front speeds) is a fundamental problem in turbulent combustion theory. A significant project is to understand how the turbulent flame speed ($s_T$) depends on the flow intensity ($A$). The G-equation is a very popular level set flame propagation model in the turbulent combustion community. The main purpose of this paper is to study properties of $\lim_{A\to +\infty} \frac{s_T}{A}$ and $\lim_{A\to +\infty}\, s_T$ (if finite, or strong bending) in the G-equation model for two-dimensional (2D) divergence-free periodic flows. Our analysis is based on the invariant measures and rotation vectors of the 2D flows and the travel times of the associated flow trajectories under control. Optimal linear/sublinear growth and strong bending conditions are precisely given in terms of rotation vectors and periodic orbits. A strong bending formula of $s_T$ in the cat's-eye flow is discovered by averaging the controlled characteristics of the G-equation. The growth rate of...

Oscillatory instability and fluid patterns in low-Prandtl-number Rayleigh-Bénard convection with uniform rotation

We present the results of direct numerical simulations of flow patterns in a low-Prandtl-number (Pr = 0.1) fluid above the onset of oscillatory convection in a Rayleigh-Bénard system rotating uniformly about a vertical axis. Simulations were carried out in a periodic box with thermally conducting and stress-free top and bottom surfaces. We considered a rectangular box (Lx × Ly × 1) and a wide range of Taylor numbers (750 ⩽ Ta ⩽ 5000) for the purpose. The horizontal aspect ratio η = Ly/Lx of the box was varied from 0.5 to 10. The primary instability appeared in the form of two-dimensional standing waves for shorter boxes (0.5 ⩽ η < 1 and 1 < η < 2). The flow patterns observed in boxes with η = 1 and η = 2 were different from those with η < 1 and 1 < η < 2. We observed a competition between two sets of mutually perpendicular rolls at the primary instability in a square cell (η = 1) for Ta < 2700, but observed a set of parallel rolls in the form of standing waves for Ta ⩾ 2700. The three-dimensional convection was temporally quasiperiodic for 750 ⩽ Ta ⩽ 1114, chaotic for 1115 ⩽ Ta < 1125, and quasiperiodic once again for 1125 ⩽ Ta < 2700. The three-dimensional quasiperiodic flow bifurcated into a two-dimensional periodic flow for Ta ⩾ 2700. The convection at the primary instability were three-dimensional and quasiperiodic in time for all values of Ta (⩽3000) in a box with η = 2. Küppers-Lortz type of convection was observed at the primary instability in longer boxes (4 ⩽ η ⩽ 10). The convective structures consisted of the appearance and disappearance of straight rolls, rhombic patterns, and wavy rolls inclined at an angle \documentclass[12pt]{minimal}\begin{document}$\phi = \frac{\pi }{2} - \arctan {(\eta ^{-1})}$\end{document}ϕ=π2−arctan(η−1) with the straight rolls. We also present a low-dimensional model constructed for a square box (η = 1).

Exact solutions of non-linear differential-difference equations of a viscous fluid with finite relaxation time

We have found a number exact of solutions to non-linear differential-difference equations of a viscous incompressible fluid with finite relaxation time. The results are used to solve a few hydrodynamic problems; in particular, we describe a one-dimensional flow arising from longitudinal periodic oscillations of a rigid plane and a two-dimensional periodic flow with pressure gradient near a fixed porous plate. We discuss issues of hydrodynamic instability of solutions as well as ways to enhance differential-difference fluid and related heat and diffusion models. The modified relaxation fluid model presented may theoretically explain one of the causes of the onset of turbulence. We also give a few exact solutions to the Racke–Saal non-linear hyperbolic fluid equations.

배플이 부착된 채널 유동의 불안정성

Flow instability is investigated in a two-dimensional channel with thin baffles placed symmetrically in the vertical direction and periodically in the streamwise dircetion. At low Reynolds numbers, the flow is steady and symmetric. Above a critical Reynolds number, the steady flow undergoes a Hopf bifurcation leading to unsteady periodic flow. As Reynolds number further increases, we observe the onset of secondary instability. At high Reynolds numbers, the two-dimensional periodic flow becomes three dimmensional. To identify the onset of secondary instability, we carry out Floquet stability analysis. We obseved the transition to 3D flow at a Reynolds number of about 125. Also, we computed dominant spanwise wavenumbers near the critical Reynolds number, and visualized vortical structures associated with the most unstable spanwise wave.

Elastic waves and transition to elastic turbulence in a two-dimensional viscoelastic Kolmogorov flow

We investigate the dynamics of the two-dimensional periodic Kolmogorov flow of a viscoelastic fluid, described by the Oldroyd-B model, by means of direct numerical simulations. Above a critical Weissenberg number the flow displays a transition from stationary to randomly fluctuating states, via periodic ones. The increasing complexity of the flow in both time and space at progressively higher values of elasticity accompanies the establishment of mixing features. The peculiar dynamical behavior observed in the simulations is found to be related to the appearance of filamental propagating patterns, which develop even in the limit of very small inertial nonlinearities, thanks to the feedback of elastic forces on the flow.

Meshless approach for wall treatment in Large-Eddy Simulation

An extended meshless approach for wall-modelling in Large-Eddy Simulation of incompressible flows that may take into account several terms of Navier–Stokes equations (the pressure gradient, the unsteady term, buoyancy effects, …) is presented. The method is based on an analytical integration of the momentum equations along the normal direction to the wall. After a detailed description of the algorithm, the present approach is applied to a two-dimensional periodic hill flow at a bulk Reynolds number of 7100. It is shown that taking into account the instantaneous pressure gradient has no effect on the mean values of the velocity. However, a substantial improvement is obtained concerning the rms values.

Floquet stability analysis of the flow around an oscillating cylinder

Abstract This investigative work is concerned with the flow around a circular cylinder submitted to forced transverse oscillations. The goal is to investigate how the transition to turbulence is initiated in the wake for cases with different Reynolds numbers ( Re ) and displacement amplitudes (A). For each Re the motion frequency is kept constant, close to the Strouhal number of the flow around a fixed cylinder at the same Re . Stability analysis of two-dimensional periodic flows around a forced-oscillating cylinder is carried out with respect to three-dimensional infinitesimal perturbations. The procedure consists of performing a Floquet type analysis of time-periodic base flows, computed using the spectral/hp element method. With the results of the Floquet calculations, considerations regarding the stability of the system are drawn, and the form of the instability at its onset is obtained. The critical Reynolds number is observed to change with the amplitude of oscillation. With respect to instabilities, unstable modes with the same symmetry as mode A of a fixed cylinder are observed; however, they present different wavelengths. Also, the instabilities observed for the oscillating cylinder are distinctively stronger in the braid shear layers. Other unstable modes similar to mode B are found. Quasi-periodic modes are observed in the 2S wake, and subharmonic mode occurrences are reported in P + S wakes.

Effects of Reynolds and Prandtl numbers on heat transfer from a square cylinder in the unsteady flow regime

The effects of the Reynolds and Prandtl numbers on the rate of heat transfer from a square cylinder are investigated numerically in the unsteady two-dimensional periodic flow regime, for the range of conditions 60 ⩽ Re ⩽ 160 and 0.7 ⩽ Pr ⩽ 50 (the maximum value of Peclet number being 4000). A semi-explicit finite volume method has been used on a non-uniform collocated grid arrangement to solve the governing equations. Using the present numerical results, simple heat transfer correlations are obtained for the constant temperature and constant heat flux conditions on the solid square cylinder. In addition, the variation of the time averaged local Nusselt number on the each face of the obstacle and representative isotherm plots are presented to elucidate the role of Prandtl number on heat transfer in the unsteady flow regime.

Stability of vortex flow in a modulated channel

Linear stability analysis of a fully developed two-dimensional periodic steady flow in a spatially modulated symmetric channel is investigated numerically. Both walls are sinusoidally modulated. The modulation amplitude is assumed to be small. The base steady flow is calculated using a regular perturbation expansion of the flow field coupled to a variable-step finite-difference scheme. The disturbance flow equations are derived within the framework of Floquet theory and solved using a two-point boundary value method. The implemented procedure is simple, direct, and very efficient. The accuracy of this method is established. The influence of geometric parameters on the threshold for the onset of instability is systematically investigated. It is established that the critical Reynolds number for the onset of hydrodynamic instability decreases as the wall modulation amplitude or wave number increases. Modulated channel flow is characterized by the formation of a recirculation region or vortices beneath the modulation crest as the Reynolds number exceeds a certain critical threshold. The relation between the thresholds for the onset of instability and vortex flow is examined over a wide range of geometric parameters. Vortex flow effects on the conditions for the onset of instability are systematically investigated. The parameter regime of stable vortex flow is established. It is found that the parameter regime of stable vortex flow increases significantly with the wall modulation wave number for small wave numbers. This increase become less pronounced as the wave number is further increased.